The presentation is covered the following topics :
1.Introduction
2.Finite Differences
(a) Forward Differences
(b) Backward Differences
(c) Central Differences
3.Interpolation for equal intervals
(a) Newton Forward and Backward Interpolation Formula
(b) Gauss Forward and Backward Interpolation Formula
(c)Stirling’s Interpolation Formula
4.Interpolation for unequal intervals
(a) Lagrange’s Interpolation Formula
5.Inverse interpolation
6.Relation between the operators
7.Newton Divided Difference Interpolation Formula
and is useful for Engineering and B.Sc students.
COM2304: Intensity Transformation and Spatial Filtering – I (Intensity Transf...Hemantha Kulathilake
At the end of this lesson, you should be able to;
describe spatial domain of the digital image.
recognize the image enhancement techniques.
describe and apply the concept of intensity transformation.
express histograms and histogram processing.
describe image noise.
characterize the types of Noise.
describe concept of image restoration.
The presentation is covered the following topics :
1.Introduction
2.Finite Differences
(a) Forward Differences
(b) Backward Differences
(c) Central Differences
3.Interpolation for equal intervals
(a) Newton Forward and Backward Interpolation Formula
(b) Gauss Forward and Backward Interpolation Formula
(c)Stirling’s Interpolation Formula
4.Interpolation for unequal intervals
(a) Lagrange’s Interpolation Formula
5.Inverse interpolation
6.Relation between the operators
7.Newton Divided Difference Interpolation Formula
and is useful for Engineering and B.Sc students.
COM2304: Intensity Transformation and Spatial Filtering – I (Intensity Transf...Hemantha Kulathilake
At the end of this lesson, you should be able to;
describe spatial domain of the digital image.
recognize the image enhancement techniques.
describe and apply the concept of intensity transformation.
express histograms and histogram processing.
describe image noise.
characterize the types of Noise.
describe concept of image restoration.
Overview on Optimization algorithms in Deep LearningKhang Pham
Overview on function optimization in general and in deep learning. The slides cover from basic algorithms like batch gradient descent, stochastic gradient descent to the state of art algorithm like Momentum, Adagrad, RMSprop, Adam.
Overview on Optimization algorithms in Deep LearningKhang Pham
Overview on function optimization in general and in deep learning. The slides cover from basic algorithms like batch gradient descent, stochastic gradient descent to the state of art algorithm like Momentum, Adagrad, RMSprop, Adam.
Deregulated Load Frequency Control (DLFC) plays an important role in power systems. The main aim of
DLFC is to minimize the deviation in area frequency and tie-line power changes. Conventional PID
controller gains are optimally tuned at one operating condition. The main problem of this controller is that
it fails to operate under different dynamic operating conditions. To overcome that drawback, fuzzy
controllers have very much importance. The design of Fuzzy controller’s mostly depends on the
Membership Functions (MF) and rule-base over the input and output ranges controllers. Many methods
were proposed to generate and minimize the fuzzy rules-base. The present paper proposes an optimal fuzzy
rule base based on Principal component analysis and the designed controller is tested on three area
deregulated interconnected thermal power system. The efficacies of the proposed controller are compared
with the Fuzzy C-Means controller and Conventional PID controller.
A LEAST ABSOLUTE APPROACH TO MULTIPLE FUZZY REGRESSION USING Tw- NORM BASED O...ijfls
A least absolute approach to multiple fuzzy regression using Tw-norm based arithmetic operations is
discussed by using the generalized Hausdorff metric and it is investigated for the crisp input- fuzzy output
data. A comparative study based on two data sets are presented using the proposed method using shape
preserving operations with other existing method.
RISK ASSESSMENT OF NATURAL HAZARDS IN NAGAPATTINAM DISTRICT USING FUZZY LOGIC...ijfls
The assessment of risks due to natural hazards is a major one responsible for risk management and the constant development of Nagapattinam district. The estimation of risk in Nagapattinam district was deduced using fuzzy logic model for the given raw data. A hierarchical fuzzy logic system with six inputs and one output is designed in Matlab software environment using fuzzy logic toolbox and simulink. The simulink investigations are done for five areas in Nagapattinam district. The fuzzy system is developed using the information sources provided by disaster management cell of Nagapattinam district.
The present work focuses on two directions. First, a new fuzzy method using triangular / trapezoidal fuzzy numbers as tools is developed for evaluating a group’s mean performance, when qualitative grades instead of numerical scores are used for assessing its members’ individual performance. Second, a new technique is applied for solving Linear Programming problems with fuzzy coefficients. Examples are presented on student and basket-ball player assessment and on real life problems involving Linear Programming under fuzzy conditions to illustrate the applicability of our results in practice. A discussion follows on the perspectives of future research on the subject and the article closes with the general conclusions.
International Refereed Journal of Engineering and Science (IRJES)irjes
The core of the vision IRJES is to disseminate new knowledge and technology for the benefit of all, ranging from academic research and professional communities to industry professionals in a range of topics in computer science and engineering. It also provides a place for high-caliber researchers, practitioners and PhD students to present ongoing research and development in these areas.
Geoid height determination is one of the major problems of geodesy because usage of satellite
techniques in geodesy isgetting increasing. Geoid heights can be determined using different methods according
to the available data. Soft computing methods such as Fuzzy logic and neural networks became so popular that
they are used to solve many engineering problems. Fuzzy logic theory and later developments in uncertainty
assessment have enabled us to develop more precise models for our requirements. In this study, How to
construct the best fuzzy model is examined. For this purpose, three different data sets were taken and two
different kinds (two inpust one output and three inputs one output) fuzzy model were formed for the calculation
of geoid heights in Istanbul (Turkey). The Fuzzy models results of these were compared with geoid heights
obtained by GPS/levelling methods. The fuzzy approximation models were tested on the test points.
MSL 5080, Methods of Analysis for Business Operations 1 .docxmadlynplamondon
MSL 5080, Methods of Analysis for Business Operations 1
Course Learning Outcomes for Unit III
Upon completion of this unit, students should be able to:
2. Distinguish between the approaches to determining probability.
3. Contrast the major differences between the normal distribution and the exponential and Poisson
distributions.
Reading Assignment
Chapter 2: Probability Concepts and Applications, pp. 32–48
Unit Lesson
Mathematical truths provide us several useful means to estimate what will happen based on factors that are
given or researched. After becoming familiar with the idea of probability, one can see how mathematics make
applications in government and business possible.
Probability Distributions
To look at probability distributions, one should define a random variable as an unknown that could be any
real number, including decimals or fractions. Many problems in life have real numbers of any value of a whole
number and fraction or decimal as the value of the random variable amount. Discrete random variables will
have a certain limited range of values, and continuous random variables may have an infinite range of
possible values. These continuous random variables could be any value at all (Render, Stair, Hanna, &
Hale, 2015).
One true tendency is that events that occur in a group of trials tend to cluster around a middle point of values
as the most occurring, or highest probabilities they will occur. They then taper off to one or both sides as there
are lower probabilities that the events will be very low from the middle (or zero) and very high from the middle.
This middle point is called the mean or expected value E(X):
n
E(X) = ∑ Xi P(Xi)
i=1
Where Xi is the random variable value, and the summation sign ∑ with n and i=1 means you are adding all n
possible values (Render et al., 2015).
The sum of these events can be shown as graphs. If the random variable has a discrete probability
distribution (e.g., cans of paint that can be sold in a day), then the graph of events may look like this:
UNIT III STUDY GUIDE
Binomial and Normal Distributions
MSL 5080, Methods of Analysis for Business Operations 2
UNIT x STUDY GUIDE
Title
The bar heights show the probability for any X (or, P(X) ) along the y-axis, given the discrete number for X
along the x-axis and no fractions for discrete variables (no half-cans of paint).
The variance (σ2) is the spread of the distribution of events in a probability distribution (Render et al., 2015).
The variance is interesting because a small variance may indicate that the event value will most likely be near
the mean most of the time, and a large variance may show that the mean is not all that reliable a guide of
what the event values will be, as the sp.
Handling missing data with expectation maximization algorithmLoc Nguyen
Expectation maximization (EM) algorithm is a powerful mathematical tool for estimating parameter of statistical models in case of incomplete data or hidden data. EM assumes that there is a relationship between hidden data and observed data, which can be a joint distribution or a mapping function. Therefore, this implies another implicit relationship between parameter estimation and data imputation. If missing data which contains missing values is considered as hidden data, it is very natural to handle missing data by EM algorithm. Handling missing data is not a new research but this report focuses on the theoretical base with detailed mathematical proofs for fulfilling missing values with EM. Besides, multinormal distribution and multinomial distribution are the two sample statistical models which are concerned to hold missing values.
A NEW APPROACH FOR RANKING SHADOWED FUZZY NUMBERS AND ITS APPLICATIONijcsit
In many decision situations, decision-makers face a kind of complex problems. In these decision-making
problems, different types of fuzzy numbers are defined and, have multiple types of membership functions.
So, we need a standard form to formulate uncertain numbers in the problem. Shadowed fuzzy numbers are
considered granule numbers which approximate different types and different forms of fuzzy numbers. In
this paper, a new ranking approach for shadowed fuzzy numbers is developed using value, ambiguity and
fuzziness for shadowed fuzzy numbers. The new ranking method has been compared with other existing
approaches through numerical examples. Also, the new method is applied to a hybrid multi-attribute
decision making problem in which the evaluations of alternatives are expressed with different types of
uncertain numbers. The comparative study for the results of different examples illustrates the reliability of
the new method.
In many decision situations, decision-makers face a kind of complex problems. In these decision-making problems, different types of fuzzy numbers are defined and, have multiple types of membership functions. So, we need a standard form to formulate uncertain numbers in the problem. Shadowed fuzzy numbers are considered granule numbers which approximate different types and different forms of fuzzy numbers. In this paper, a new ranking approach for shadowed fuzzy numbers is developed using value, ambiguity and fuzziness for shadowed fuzzy numbers. The new ranking method has been compared with other existing approaches through numerical examples. Also, the new method is applied to a hybrid multi-attribute decision making problem in which the evaluations of alternatives are expressed with different types of uncertain numbers. The comparative study for the results of different examples illustrates the reliability of the new method.
A New Approach for Ranking Shadowed Fuzzy Numbers and its Application IJCSITJournal2
n many decision situations, decision-makers face a kind of complex problems. In these decision-making
problems, different types of fuzzy numbers are defined and, have multiple types of membership functions.
So, we need a standard form to formulate uncertain numbers in the problem. Shadowed fuzzy numbers are
considered granule numbers which approximate different types and different forms of fuzzy numbers. In
this paper, a new ranking approach for shadowed fuzzy numbers is developed using value, ambiguity and
fuzziness for shadowed fuzzy numbers. The new ranking method has been compared with other existing
approaches through numerical examples. Also, the new method is applied to a hybrid multi-attribute
decision making problem in which the evaluations of alternatives are expressed with different types of
uncertain numbers. The comparative study for the results of different examples illustrates the reliability of
the new method.
In this paper, the L1 norm of continuous functions and corresponding continuous estimation of regression parameters are defined. The continuous L1 norm estimation problem of one and two parameters linear models in the continuous case is solved. We proceed to use the functional form and parameters of the probability distribution function of income to exactly determine the L1 norm approximation of the corresponding Lorenz curve of the statistical population under consideration.
Covid19py by Konstantinos Kamaropoulos
A tiny Python package for easy access to up-to-date Coronavirus (COVID-19, SARS-CoV-2) cases data.
ref:https://github.com/Kamaropoulos/COVID19Py
https://pypi.org/project/COVID19Py/?fbclid=IwAR0zFKe_1Y6Nm0ak1n0W1ucFZcVT4VBWEP4LOFHJP-DgoL32kx3JCCxkGLQ
"optrees" package in R and examples.(optrees:finds optimal trees in weighted ...Dr. Volkan OBAN
Finds optimal trees in weighted graphs. In
particular, this package provides solving tools for minimum cost spanning
tree problems, minimum cost arborescence problems, shortest path tree
problems and minimum cut tree problem.
by Volkan OBAN
k-means Clustering in Python
scikit-learn--Machine Learning in Python
from sklearn.cluster import KMeans
k-means clustering is a method of vector quantization, originally from signal processing, that is popular for cluster analysis in data mining. k-means clustering aims to partition n observations into k clusters in which each observation belongs to the cluster with the nearest mean, serving as a prototype of the cluster. This results in a partitioning of the data space into Voronoi cells.
The problem is computationally difficult (NP-hard); however, there are efficient heuristic algorithms that are commonly employed and converge quickly to a local optimum. These are usually similar to the expectation-maximization algorithm for mixtures of Gaussian distributions via an iterative refinement approach employed by both algorithms. Additionally, they both use cluster centers to model the data; however, k-means clustering tends to find clusters of comparable spatial extent, while the expectation-maximization mechanism allows clusters to have different shapes.[wikipedia]
ref: http://scikit-learn.org/stable/auto_examples/cluster/plot_cluster_iris.html
Forecasting through ARIMA Modeling using R
ref:http://ucanalytics.com/blogs/step-by-step-graphic-guide-to-forecasting-through-arima-modeling-in-r-manufacturing-case-study-example/
k-means Clustering and Custergram with R.
K Means Clustering is an unsupervised learning algorithm that tries to cluster data based on their similarity. Unsupervised learning means that there is no outcome to be predicted, and the algorithm just tries to find patterns in the data. In k means clustering, we have the specify the number of clusters we want the data to be grouped into. The algorithm randomly assigns each observation to a cluster, and finds the centroid of each cluster.
ref:https://www.r-bloggers.com/k-means-clustering-in-r/
ref:https://rpubs.com/FelipeRego/K-Means-Clustering
ref:https://www.r-bloggers.com/clustergram-visualization-and-diagnostics-for-cluster-analysis-r-code/
Data Science and its Relationship to Big Data and Data-Driven Decision MakingDr. Volkan OBAN
Data Science and its Relationship to Big Data and Data-Driven Decision Making
To cite this article:
Foster Provost and Tom Fawcett. Big Data. February 2013, 1(1): 51-59. doi:10.1089/big.2013.1508.
Foster Provost and Tom Fawcett
Published in Volume: 1 Issue 1: February 13, 2013
ref:http://online.liebertpub.com/doi/full/10.1089/big.2013.1508
https://www.researchgate.net/publication/256439081_Data_Science_and_Its_Relationship_to_Big_Data_and_Data-Driven_Decision_Making
R Machine Learning packages( generally used)
prepared by Volkan OBAN
reference:
https://github.com/josephmisiti/awesome-machine-learning#r-general-purpose
Seminar of U.V. Spectroscopy by SAMIR PANDASAMIR PANDA
Spectroscopy is a branch of science dealing the study of interaction of electromagnetic radiation with matter.
Ultraviolet-visible spectroscopy refers to absorption spectroscopy or reflect spectroscopy in the UV-VIS spectral region.
Ultraviolet-visible spectroscopy is an analytical method that can measure the amount of light received by the analyte.
Cancer cell metabolism: special Reference to Lactate PathwayAADYARAJPANDEY1
Normal Cell Metabolism:
Cellular respiration describes the series of steps that cells use to break down sugar and other chemicals to get the energy we need to function.
Energy is stored in the bonds of glucose and when glucose is broken down, much of that energy is released.
Cell utilize energy in the form of ATP.
The first step of respiration is called glycolysis. In a series of steps, glycolysis breaks glucose into two smaller molecules - a chemical called pyruvate. A small amount of ATP is formed during this process.
Most healthy cells continue the breakdown in a second process, called the Kreb's cycle. The Kreb's cycle allows cells to “burn” the pyruvates made in glycolysis to get more ATP.
The last step in the breakdown of glucose is called oxidative phosphorylation (Ox-Phos).
It takes place in specialized cell structures called mitochondria. This process produces a large amount of ATP. Importantly, cells need oxygen to complete oxidative phosphorylation.
If a cell completes only glycolysis, only 2 molecules of ATP are made per glucose. However, if the cell completes the entire respiration process (glycolysis - Kreb's - oxidative phosphorylation), about 36 molecules of ATP are created, giving it much more energy to use.
IN CANCER CELL:
Unlike healthy cells that "burn" the entire molecule of sugar to capture a large amount of energy as ATP, cancer cells are wasteful.
Cancer cells only partially break down sugar molecules. They overuse the first step of respiration, glycolysis. They frequently do not complete the second step, oxidative phosphorylation.
This results in only 2 molecules of ATP per each glucose molecule instead of the 36 or so ATPs healthy cells gain. As a result, cancer cells need to use a lot more sugar molecules to get enough energy to survive.
Unlike healthy cells that "burn" the entire molecule of sugar to capture a large amount of energy as ATP, cancer cells are wasteful.
Cancer cells only partially break down sugar molecules. They overuse the first step of respiration, glycolysis. They frequently do not complete the second step, oxidative phosphorylation.
This results in only 2 molecules of ATP per each glucose molecule instead of the 36 or so ATPs healthy cells gain. As a result, cancer cells need to use a lot more sugar molecules to get enough energy to survive.
introduction to WARBERG PHENOMENA:
WARBURG EFFECT Usually, cancer cells are highly glycolytic (glucose addiction) and take up more glucose than do normal cells from outside.
Otto Heinrich Warburg (; 8 October 1883 – 1 August 1970) In 1931 was awarded the Nobel Prize in Physiology for his "discovery of the nature and mode of action of the respiratory enzyme.
WARNBURG EFFECT : cancer cells under aerobic (well-oxygenated) conditions to metabolize glucose to lactate (aerobic glycolysis) is known as the Warburg effect. Warburg made the observation that tumor slices consume glucose and secrete lactate at a higher rate than normal tissues.
Professional air quality monitoring systems provide immediate, on-site data for analysis, compliance, and decision-making.
Monitor common gases, weather parameters, particulates.
Observation of Io’s Resurfacing via Plume Deposition Using Ground-based Adapt...Sérgio Sacani
Since volcanic activity was first discovered on Io from Voyager images in 1979, changes
on Io’s surface have been monitored from both spacecraft and ground-based telescopes.
Here, we present the highest spatial resolution images of Io ever obtained from a groundbased telescope. These images, acquired by the SHARK-VIS instrument on the Large
Binocular Telescope, show evidence of a major resurfacing event on Io’s trailing hemisphere. When compared to the most recent spacecraft images, the SHARK-VIS images
show that a plume deposit from a powerful eruption at Pillan Patera has covered part
of the long-lived Pele plume deposit. Although this type of resurfacing event may be common on Io, few have been detected due to the rarity of spacecraft visits and the previously low spatial resolution available from Earth-based telescopes. The SHARK-VIS instrument ushers in a new era of high resolution imaging of Io’s surface using adaptive
optics at visible wavelengths.
Comparing Evolved Extractive Text Summary Scores of Bidirectional Encoder Rep...University of Maribor
Slides from:
11th International Conference on Electrical, Electronics and Computer Engineering (IcETRAN), Niš, 3-6 June 2024
Track: Artificial Intelligence
https://www.etran.rs/2024/en/home-english/
Richard's aventures in two entangled wonderlandsRichard Gill
Since the loophole-free Bell experiments of 2020 and the Nobel prizes in physics of 2022, critics of Bell's work have retreated to the fortress of super-determinism. Now, super-determinism is a derogatory word - it just means "determinism". Palmer, Hance and Hossenfelder argue that quantum mechanics and determinism are not incompatible, using a sophisticated mathematical construction based on a subtle thinning of allowed states and measurements in quantum mechanics, such that what is left appears to make Bell's argument fail, without altering the empirical predictions of quantum mechanics. I think however that it is a smoke screen, and the slogan "lost in math" comes to my mind. I will discuss some other recent disproofs of Bell's theorem using the language of causality based on causal graphs. Causal thinking is also central to law and justice. I will mention surprising connections to my work on serial killer nurse cases, in particular the Dutch case of Lucia de Berk and the current UK case of Lucy Letby.
A brief information about the SCOP protein database used in bioinformatics.
The Structural Classification of Proteins (SCOP) database is a comprehensive and authoritative resource for the structural and evolutionary relationships of proteins. It provides a detailed and curated classification of protein structures, grouping them into families, superfamilies, and folds based on their structural and sequence similarities.
THE IMPORTANCE OF MARTIAN ATMOSPHERE SAMPLE RETURN.Sérgio Sacani
The return of a sample of near-surface atmosphere from Mars would facilitate answers to several first-order science questions surrounding the formation and evolution of the planet. One of the important aspects of terrestrial planet formation in general is the role that primary atmospheres played in influencing the chemistry and structure of the planets and their antecedents. Studies of the martian atmosphere can be used to investigate the role of a primary atmosphere in its history. Atmosphere samples would also inform our understanding of the near-surface chemistry of the planet, and ultimately the prospects for life. High-precision isotopic analyses of constituent gases are needed to address these questions, requiring that the analyses are made on returned samples rather than in situ.
What is greenhouse gasses and how many gasses are there to affect the Earth.moosaasad1975
What are greenhouse gasses how they affect the earth and its environment what is the future of the environment and earth how the weather and the climate effects.
2. 1 Introduction
Recent articles, such as McCauley-Bell et al. (1999) and Sánchez and Gómez (2003a,
2003b, 2004), used fuzzy regression (FR) in their analysis. The former use it to predict
the relationship of known risk factors to the onset of occupational injury, while the latter
used it to investigate the term structure of interest rates (TSIR). Following Tanaka et. al.
(1982), their models took the general form:
nn xAxAAY
~~~~
110 +++= L (1)
where Y
~
is the fuzzy output, Ãi, j=1,2,..., n, is a fuzzy coefficient, and x = (x1, ..., xn) is an
n-dimensional non-fuzzy input vector. The fuzzy components were assumed to be
triangular fuzzy numbers (TFNs). Consequently, the coefficients, for example, can be
characterized by a membership function (MF), µA(a), a representation of which is shown
in Figure 1.
Figure 1: Fuzzy Coefficient
As indicated, the salient features of the TFN are its mode, its left and right spread, and its
support. When the two spreads are equal, the TFN is known as a symmetrical TFN
(STFN).
The basic idea of the Tanaka approach, often referred to as possibilistic regression, was to
minimize the fuzziness of the model by minimizing the total spread of the fuzzy
coefficients, subject to including all the given data.
The purpose of this article is to revisit the fuzzy regression portions of the foregoing
studies, and to discuss issues related to the Tanaka (possibilistic) regression model. This
ARC2005_Shapiro_06.pdf 2
3. discussion is not meant to be exhaustive but, rather, is intended to point out some of the
major considerations. The outline of the paper is as follows. We first define and
conceptualize the general components of fuzzy regression. Next, the essence of the
Tanaka model is explored, including a commentary on some of its potential limitations.
Then, fuzzy least-squares regression models are discussed as an alternative to the Tanaka
model. Throughout the paper, the same simple data set is used to show how the ideas are
implemented. The paper ends with a summary of the conclusions of the study.
2 Fuzzy Linear Regression Basics
This section provides an introduction to fuzzy linear regression. The topics addressed
include the motivation for FR, the components of FR, fuzzy coefficients, the h-certain
factor, and fuzzy output.
2.1 Motivation
Classical statistical linear regressions takes the form
(2)mixxy iikkii ,...,2,1,110 =++++= εβββ L
where the dependent (response) variable, yi , the independent (explanatory) variables, xij,
and the coefficients (parameters), βj, are crisp values, and εi is a crisp random error term
with E(εi)=0, variance σ2
(εi )=σ2
, and covariance σ(εi , εj) = 0, ∀i,j, i≠ j.
Although statistical regression has many applications, problems can occur in the
following situations:
• Number of observations is inadequate (Small data set)
• Difficulties verifying distribution assumptions
• Vagueness in the relationship between input and output variables
• Ambiguity of events or degree to which they occur
• Inaccuracy and distortion introduced by linearalization
Thus, statistical regression is problematic if the data set is too small, or there is difficulty
verifying that the error is normally distributed, or if there is vagueness in the relationship
between the independent and dependent variables, or if there is ambiguity associated with
the event or if the linearity assumption is inappropriate. These are the very situations
fuzzy regression was meant to address.
2.2 The Components of Fuzzy Regression
There are two general ways (not necessarily mutually exclusive) to develop a fuzzy
regression model: (1) models where the relationship of the variables is fuzzy; and (2)
ARC2005_Shapiro_06.pdf 3
4. models where the variables themselves are fuzzy. Both of these models are explored in
the rest of this article, but, for this conceptualization, we focus on models where the data
is crisp and the relationship of the variables is fuzzy.
It is a simple matter to conceptualize fuzzy regression. Consider for this, and subsequent,
examples the following simple Ishibuchi (1992) data:
Table 1: Data Pairs
i 1 2 3 4 5 6 7 8
xi 2 4 6 8 10 12 14 16
yi 14 16 14 18 18 22 18 22
Starting with this data, we fit a straight line through two or more data points in such a
way that it bounds the data points from above. Here, these points are determined
heuristically and OLS is used to compute the parameters of the line labeled YH
, which
takes the values , as shown in Figure 2(a).xy 75.13ˆ +=
Figure 2: Conceptualizing the upper and lower bound
Similarly, we fit a second straight line through two or more data points in such a way that
it bounds the data points from below. As shown in Figure 2(b), the fitted line in
this case is labeled YL
and takes the values xy 5.11ˆ += .
Assuming, for the purpose of this example, that STFN are used for the MFs, the modes of
the MFs fall midway between the boundary lines.1
ARC2005_Shapiro_06.pdf 4
1
This approach to choosing the mode was discussed by Wang and Tsaur (2000) p. 357.
5. For any given data pair, (xi, yi), the foregoing conceptualizations can be summarized by
the fuzzy regression interval [Y shown in Figure 3.]Y, U
i
L
i
2
Figure 3: Fuzzy Regression Interval
1h
iY =
is the mode of the MF and if a SFTN is assumed, )/2Y(YY L
i
U
ii
1h
i +===
)Y,Y, 1h
i
L
i
U
i
=
L
iY
Y . Given
the parameters, (YU
,YL
, Yh=1
), which characterize the fuzzy regression model, the i-th
data pair (xi,yi), is associated with the model parameters (Y . From a
regression perspective, we can view - yU
iY
U
iY -
i and yi - as components of the SST, yL
iY
1h
i
=
i -
as a component of SSE, and and - as components of the SSR, as
discussed by Wang and Tsaur (2000).
1h
iY = 1h
iY =
Y
In possibilistic regression based on STFN, only the data points involved in determining
the upper and lower bounds determine the structure of the model, as depicted in Figure 2.
The rest of the data points have no impact on the structure. This problem is resolved by
using asymmetric TFNs.
2.3 The Fuzzy Coefficients
Combining Equation (1) and Figure 1, and, for the present, restricting the discussion to
STFNs, the MF of the j-th coefficient, may be defined as:
−
−= 0,
||
1max)(
j
j
A
c
aa
aj
µ (3)
where aj is the mode and cj is the spread, and represented as shown in Figure 4.
ARC2005_Shapiro_06.pdf 5
2
Adapted from Wang and Tsaur (2000), Figure 1.
6. Figure 4: Symmetrical fuzzy parameters
Defining
(
{ } { } njcaAcaAcaA LjjjjjjLjjj ,,1,0,
~
:
~
,
~
L=+≤≤−== (4)
and restricting consideration to the case where only the coefficients are fuzzy, we can
write
5)
∑=
+=
n
j
iji xAAY
1
10
~~~
∑=
+=
n
j
ijLjjL xcaca
1
00 ),(),(
This is a useful formulation because it explicitly portrays the mode and spreads of the
fuzzy parameters. In a subsequent section, we explore fuzzy independent variables.
2.4 The "h-certain" Factor
If, as in Figure 3, the supports3
are just sufficient to include all the data points of the
sample, there would be only limited confidence in out-of-sample projection using the
estimated FR model. This is resolved for FR, just as it is with statistical regression, by
extending the supports.
Consider the MF associated with the j-th fuzzy coefficient, a representation of which is
shown in Figure 5.
ARC2005_Shapiro_06.pdf 6
3
Support functions are discussed in Diamond (1988: 143) and Wünsche and Näther (2002: 47).
7. Figure 5: Estimating Aj using an "h-certain" factor
For illustrative purposes, a non-symmetric TFN is shown, wherec andc represent the
left and right spread respectively. Beyond that, what makes this MF materially different
from the one shown in Figure 4, is that it contains a point "h" on the y-axis, called an "h-
certain factor," which, by controlling the size of the feasible data interval (the base of the
shaded area), extends the support of the MF.
L
j
R
j
4
In particular, as the h-factor increases for a
given data set, so increases the spreads,c and .L
j
R
jc
2.5 Observed Fuzzy Output
An h-certain factor also can be applied to the observed output. Thus, the i-th output data
might be represented by the STFN, )e,(yY
~
iii = , where yi is the mode and ei is the spread,
as shown in Figure 6. Here, the actual data points fall within the interval yi ± (1-h) ei, the
base of the shaded portion of the graph.
ARC2005_Shapiro_06.pdf 7
4
Note that the h-factor has the opposite purpose of an α-cut, in that the former is used to extend the
support, while the latter is used to reduce the support.
8. Figure 6: Observed Fuzzy Output
2.6 Fitting the Fuzzy Regression Model
Given the foregoing, two general approaches are used to fit the fuzzy regression model:
The possibilistic model. Minimize the fuzziness of the model by minimizing the total
spreads of its fuzzy coefficients (see Figure 1), subject to including the data points of
each sample within a specified feasible data interval.
The least-squares model. Minimize the distance between the output of the model and
the observed output, based on their modes and spreads.
The details of these approaches are addressed in the next two sections of this paper.
3 The Possibilistic Regression Model
The possibilistic regression model is optimized by minimizing the spread, subject to
adequate containment of the data. The spread is minimized
0c,|x|ccmin j
n
1j
ijj0 ≥
+ ∑=
(6)
Figure 7 shows the first step in the containment requirement, by showing how Figure 5
can be easily extended to portray the fuzzy output of the model.
ARC2005_Shapiro_06.pdf 8
9. Figure 7: Fuzzy output of the model
Putting this together with the observed fuzzy output, Figure 6, results in Figure 8, which
shows a representation of how the estimated fuzzy output may be fitted to the observed
fuzzy data.
Figure 8: Fitting the estimated output to the observed output
The key is that the observed fuzzy data, adjusted for the h-certain factor, is contained
within the estimated fuzzy output, adjusted for the h-certain factor. Formally,
ii
n
j
n
j
ijjijj ehyxcchxaa )1(||)1(
1 1
00 −+>
+−++∑ ∑= =
(7)
ARC2005_Shapiro_06.pdf 9
10. ii
n
j
n
j
ijjijj ehyxcchxaa )1(||)1(
1 1
00 −−<
+−−+∑ ∑= =
cj $0, i = 0, 1, ..., m, j = 0, 1, ..., n
Figure 95
shows the impact of the h-factor on the sample data, given h=0 and h=.7.
Figure 9: FLR and h-certain model
The result is what one would expect. Increasing the h-factor expands the confidence
interval and, thus, increases the probability that out-of-sample values will fall within the
model. This is comparable to increasing the confidence in statistical regression by
increasing the confidence interval.
The possibilistic linear regression model, as depicted by equations (6) and (7), is
essentially the fuzzy regression model used by Sánchez and Gómez (2003a, 2003b, 2004)
to investigate the TSIR.6
5
Adapted from Chang and Ayyub (2001), Figure 4.
6
Key components of the Sánchez and Gómez methodology included constructing a discount function from
a linear combination of quadratic or cubic splines, the coefficients of which were assumed to be TFNs or
STFNs, and using the minimum and maximum negotiated price of fixed income assets to obtain the spreads
of the dependent variable observations. Given the fuzzy discount functions, the authors provided TFN
approximations for the corresponding spot rates and forward rates. It was necessary to approximate the
spot rates and forward rates since they are nonlinear functions of the discount function, and hence are not
TFNs even though the discount function is a TFN.
ARC2005_Shapiro_06.pdf 10
11. 3.1 Criticisms of the Possibilistic Regression Model
There are a number of criticisms of the possibilistic regression model. Some of the major
ones are the following:
• Tanaka et al "used linear programming techniques to develop a model superficially
resembling linear regression, but it is unclear what the relation is to a least-squares
concept, or that any measure of best fit by residuals is present." [Diamond (1988:
141-2)]
• The original Tanaka model was extremely sensitive to the outliers. [Peters (1994)].
• There is no proper interpretation about the fuzzy regression interval [Wang and
Tsaur (2000)]
• Issue of forecasting have to be addressed [Savic and Pedrycz (1991)]
• The fuzzy linear regression may tend to become multicollinear as more independent
variables are collected [Kim et al (1996)].
• The solution is xj point-of-reference dependent, in the sense that the predicted
function will be very different if we first subtract the mean of the independent
variables, using (xj - ix ) instead of xj. [Hojati (2004), Bardossy (1990) and Bardossy
et al (1990)]
4 The Fuzzy Least-Squares Regression (FLSR) Model
An obvious way to bring the FR more in line with statistical regression is to model the
fuzzy regression along the same lines. In the case of a single explanatory variable, we
start with the standard linear regression model: [Kao and Chyu (2003)]
(8)m1,2,...,i,εxββy ii10i =++=
which in a comparable fuzzy model might take the form:
m1,2,...,i,ε~X
~
ββY
~
ii10i =++= (9)
Conceptually, the relationship between the fuzzy i-th response and explanatory variables
in (9) can be represented as shown in Figure 10.
ARC2005_Shapiro_06.pdf 11
12. Figure 10: Fuzzy i-th response and explanatory variables
Rearranging the terms in (9),
m1,2,...,i,X
~
ββY
~
ε~
i10ii =−−= (10)
From a least-squares perspective, the problem then becomes
2
10
1
)
~~
(min i
n
i
i XbbY −−∑
=
(11)
There are a number of ways to implement FLSR, but the two basic approaches are FLSR
using distance measures and FLSR using compatibility measures. A description of these
methods follows.
4.1 FLSR using Distance Measures (Diamond's Approach)
Diamond (1988) was the first to implement the FLSR using distance measures and his
methodology is the most commonly used. Essentially, he defined an L2
- metric d(.,.)2
between two TFNs by [Diamond (1988: 143) equation (2)]
(12)( ) ( )
( )2
2211
2
2211
2
21
2
222111
)()(
)()()(,,,,,
rmrm
lmlmmmrlmrlmd
+−++
−−−+−=
Given TFNs, it provides a measure of the distance between two fuzzy numbers based on
their modes, left spread and right spread.7
7
The methods of Diamond's paper are rigorously justified by a projection-type theorem for cones on a
Banach space containing the cone of triangular fuzzy numbers, where a Banach space is a normed vector
space that is complete as a metric space under the metric d(x, y) = ||x-y|| induced by the norm.
ARC2005_Shapiro_06.pdf 12
13. The case most similar to the Sánchez and Gómez model takes the form
mixY iii ,...,2,1,~~~~
10 =++= εββ (13)
and requires the optimization of
2
,
)
~
,
~~
(min ii
BA
YxBAd∑ + (14)
The solution follows from (12), and if B
~
is positive, it takes the form:
222
)()()
~
,
~~
( L
Yii
L
B
L
Aiiiii i
cyxccbxaybxaYBxAd +−−−++−+=+
(15)
A similar expression holds when
2
)( R
Yii
R
B
R
Ai i
cyxccbxa +−++++
B
~
is negative. If the solutions exist, the parameters of
A
~
and B
~
course, this fitted model has the same general characteristics as previously shown, but
now we can use the residual sum of d-squares to gauge the effectiveness of model.
In the case most reminiscent of statistical regression, the coefficients are crisp and the
task becomes the least-squares optimization problem
satisfy a system of six equations in the same number of unknowns, these
quations arising from the derivatives associated with (15) being set equal to zero. Of
(16)
Once again, the solution is gi to account t n of b.
Finally, an interesting problem enting the Diamond approach is associated
ith models of the form
ral solution, since the LHS,
e
,
)
~
,
~
(min ii
ba
YXbad∑ + 2
ven by (12), adjusted to take
when implem
in he sig
w
(17)
for which there is no gene
miXY iii ,...,2,1,10 =++= εββ ~~~~~
iY
~
One approach to this problem (Hong et al (2001)) is to replace the t-norm min(a,b)
the t-norm Tw(a,b) = a, if b=1; b, if a=1; 0, otherwise. Since T
, is a TF while the RHS
involves the fuzzy product
N
iX
~~
1β , whose sides are drumlike.
with
w(a,b) is a shape preserving
peration under multiplication, it resolves the problem. This approach is used in Koissio
and Shapiro (2005).
Another approach is to use approximate TFNs. This was done by Sánchez and Gómez
(2003a), albeit in another context.
ARC2005_Shapiro_06.pdf 13
14. 4.2 FLSR using compatibility measures
An alternate least-squares approach is based on the Celmiņš (1987) compatibility
(18)
As indicated, when the
odes of the MFs coincide.
elmiņš compatibility model, which involved maximizing the compatibility between the
(19)
Thus, for example, when there is a single crisp expla
(2001: 190)]
(20)
m1 are determined using weighted LS regression, and c0, c1, and c01 are
determined using iteration and the desired compatibility measure.
measure
(),(min{max)
~
,
~
( XBA µµγ =
representative examples of which are shown in Figure 11.8
Figure 11: Celmiņš Compatibility Measure
γ ranges from 0, when the MFs are mutually exclusive, to 1,
m
C
data and the fitted model, follows from this measure. The objective function is
natory variable, [Chang and Ayyub
1=
−
m
iγ
i
)}XBA
x
2
)1(∑
22
101010
10
2
~~ˆ
xcxccxmm
xAAY
++±+=
+=
where m0 and
8
Adapted from Chang and Ayyub (2001), Figure 2.
ARC2005_Shapiro_06.pdf 14
15. An example of the use of the Celmiņš compatibility model applied to our sample data is
own in Figure 12.9
The essential characte rves for the
e broader the width
of the bounds.
cCauley-Bell et al. (1999) and Sánchez and Gómez (2003a, 2003b,
004) provide some interesting insights into the use of fuzzy regression. However, their
ethodology relies on possibilitic regression, which has the potential limitations
1. Since some of these limitations can be circumvented by using
mportant that researchers are familiar with these techniques as
ell. If this article helps in this regard, it will have served its purpose.
hydrology,"
Water Resources Research 26, 1497-1508.
sh
Figure 12: FLS using maximum compatibility criterion
ristics of the model in this case are the parabolic cu
upper and lower bounds and that the higher the compatibility level, th
5 Comment
The studies of M
2
m
mentioned in section 3.
FLSR techniques, it is i
w
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